Singular Metrics Research Articles

Evolving window factor analysis-multivariate curve resolution with automated library matching for enhanced peak deconvolution in gas chromatography-mass spectrometry fuel data

Fuel chromatography is inherently limited by the high complexity of petroleum fuel compositions. In practice, almost no fuel components are fully resolved in gas chromatography. This is due to both insufficient peak capacity for the large number of individual components within time and chromatographic efficiency constraints, and insufficient resolving power of the stationary phase in the gas chromatography column relative to the many structurally similar isomers or homologs present in petrochemical fuels. Multidimensional approaches, longer columns and slower heating rates can offer some benefits but will not necessarily fully resolve co-eluting fuel compounds, especially within reasonable analysis times. The following work details how deconvolved mass spectral loadings, combined with library matching, provide a quality metric against which to automatically evaluate results obtained from an experimental evolving window factor analysis-multivariate curve resolution deconvolution algorithm applied to gas chromatography-mass spectrometry data. This algorithm was evaluated in the context of trace component detection in synthetic fuel data sets, dodecane and tetradecane detection in petrochemical fuels, and the detection of natural products unlikely to be present in petrochemical fuels. In the case of the trace component detection challenge, the experimental algorithm outperformed a control algorithm that utilized a singular value-based quality metric. Meanwhile, when detecting dodecane, tetradecane, and natural products in petrochemical fuels, the experimental algorithm allowed for higher-quality compound identification results than could be obtained without peak deconvolution, thus reliably improving fuel component resolution in an automated fashion.

An optimal $L^2$ extension theorem on weakly pseudoconvex Kähler manifolds

In this paper, we prove an $L^2$ extension theorem for holomorphic sections of holomorphic line bundles equipped with singular metrics on weakly pseudoconvex Kahler manifolds. Furthermore, in our $L^2$ estimate, optimal constants corresponding to variable denominators are obtained. As applications, we prove an $L^q$ extension theorem with an optimal estimate on weakly pseudoconvex Kahler manifolds and the log-plurisubharmonicity of the fiberwise Bergman kernel in the Kahler case.

Geomorphometry in Landscape Ecology: Issues of Scale, Physiography, and Application

Topographic measures are frequently used in a variety of landscape ecology applications, in their simplest form as elevation, slope, and aspect, but increasingly more complex measures are being employed. We explore terrain metric similarity with changes in scale, both grain and extent, and examine how selecting the best measures is sensitive to changes in application. There are three types of topographic measures: 1) those that relate to orientation for approximating solar input, 2) those that capture variability in terrain configuration, and 3) those that provide metrics about landform features. Many biodiversity hotspots and predators have been found to coincide with areas of complexity, yet most complexity measures cannot differentiate between terrain steepness and uneven and broken terrain. Currently characterizing terrain in landscape-level analyses can be challenging, especially at coarser spatial resolutions but developing methods that improve landscape-level assessments include multivariate approaches and the use of neighborhood statistics. Some measures are sensitive to the spatial grain of calculation, the physiography of the landscape, and the scale of application. We show which measures have the potential to be multi-collinear, and illustrate with a case study how the selection of the best measures can change depending on the question at hand using mountain lion (Puma concolor) occurrence data. The case study showed a combination of infrequently employed metrics, such as view-shed analysis and focal statistics, outperform more commonly employed singular metrics. The use of focal statistics as a measure of topographic complexity shows promise for improving how mountain lions use terrain features.

Some remarks on Einstein–Randers metrics

In this essay, we study the sufficient and necessary conditions for a Randers metric F=α+β (α is a Riemann metric, β is a 1-form) to be of constant Ricci curvature, without the restriction of strong convexity (regularity). A classification result for the case ‖β‖α>1 is provided, which is similar to the famous Bao–Robles–Shen's result for strongly convex Randers metrics (‖β‖α<1). Based on some famous vacuum Einstein metrics in General Relativity, many non-regular Einstein–Randers metrics are constructed. Besides, we find that the case ‖β‖α≡1 is very distinctive. These metrics will be called singular Randers metrics or parabolic Finsler metrics since their indicatrixs are parabolic hypersurfaces. A preliminary discussion for such metrics is provided.

Chern forms of singular metrics on vector bundles

We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson–Păun. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.

Modeling lean maintenance metric using incidence matrix approach

Lean Maintenance (LM) enhances organizational profitability by identifying and eliminating maintenance related wastes. There exists no singular metric that measures maintenance related wastes. The paper identifies the LM features and models them using incidence-matrix. LM features are represented by diagonal elements, while its off-diagonal elements represent mutual influences among the LM features. The maintenance system leanness is quantified using the permanent of the matrix. The metric of leanness is proposed to be defined as Lean maintenance index (LMI) and is a ratio of the actual to the ideal values of permanent of actual and ideal maintenance system matrices. A high value of LMI indicates that the maintenance system is operating in a reduced waste scenario with respect to its resources. Among all the LM features, LMI was found to be most sensitive to management support including organizational processes. The results of the methodology are a good guide for managers. The shortcoming of the methodology is that, it relies on values and weights of the inter-relations among the features, which may not be necessarily true and may need further scientific rigor. The proposed methodology of using LMI as a singular metric to judge maintenance efficacy is expected to aid the operation managers in quantifying the maintenance leanness and may help them focus their efforts appropriately. There is no evidence to indicate existence of comprehensive list of LM features that culminate into a singular metric of maintenance productivity. This paper attempts to fill this gap.

Energy-Efficient Cross-Layer Design of Wireless Mesh Networks for Content Sharing in Online Social Networks

Bearing in mind the associated security and privacy concerns, users in an online social network (OSN) normally communicate with their direct friends. We propose an energy-efficient cross-layer design for wireless mesh network (WMN) aided content sharing in OSN as a case study of the interplay between OSNs and technological networks. A power control aided WMN is proposed for energy-efficient content sharing. Based on our power control scheme in the physical layer, an interference-constrained-channel-reuse scheme and a Dijkstra algorithm aided energy-efficient routing protocol are invoked for the medium access control layer and for the network layer, respectively. Inspired by the tool of social network analysis, both the singular and composite betweenness metrics are invoked for quantifying the duty-cycle of mesh routers. The former only considers the topology of the mesh backbone, while the latter considers the topologies of both the OSN and the WMN. Furthermore, the singular/composite betweenness is exploited for designing sophisticated caching strategies in order to improve the content sharing performance. We demonstrate that our caching strategies are capable of reducing the total energy dissipation by as much as ${\text{40}}\%$ . Specifically, the composite betweenness based caching strategy is capable of reducing the energy dissipation of the mesh clients by ${\text{358}}\%$ , compared to its singular betweenness based counterpart.

Radiation risk index for pediatric CT: a patient-derived metric

There is a benefit in characterizing radiation-induced cancer risk in pediatric chest and abdominopelvic CT: a singular metric that represents the whole-body radiation burden while also accounting for age, gender and organ sensitivity. To compute an index of radiation risk for pediatric chest and abdominopelvic CT. Using a protocol approved by our institutional review board, 42 pediatric patients (age: 0-16years, weight: 2-80kg) were modeled into virtual whole-body anatomical models. Organ doses were estimated for clinical chest and abdominopelvic CT examinations of the patients using validated Monte Carlo simulations of two major scanner models. Using age-, size- and gender-specific organ risk coefficients, the values were converted to normalized effective dose (by dose length product) (denoted as the k factor) and a normalized risk index (denoted as the q factor). An analysis was performed to determine how these factors are correlated with patient age and size for both males and females to provide a strategy to better characterize individualized risk. The k factor was found to be exponentially correlated with the average patient diameter. For both genders, the q factor also exhibited an exponential relationship with both the average patient diameter and with patient age. For both factors, the differences between the scanner models were less than 8%. The study defines a whole-body radiation risk index for chest and abdominopelvic CT imaging, that incorporates individual estimated organ dose values, organ radiation sensitivity, patient size, exposure age and patient gender. This indexing metrology enables the assessment and potential improvement of chest and abdominopelvic CT performance through surveillance of practice dose profiles across patients and may afford improved informed communication.

An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities

The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain L 2 L^{2} -estimates of solutions of the ∂ ¯ \overline {\partial } -equation by using the de Rham-Weil isomorphism between the ∂ ¯ \overline {\partial } -cohomology and the C ˇ \rm {\check {C}} ech cohomology.

On Singular Square Metrics with Vanishing Douglas Curvature

Square metrics \(F=\frac{(\alpha +\beta )^2}{\alpha }\) are a special class of Finsler metrics. It is the rate kind of metric category to be of excellent geometrical properties. In this paper, we discuss the so-called singular square metrics \(F=\frac{(b\alpha +\beta )^2}{\alpha }\). A characterization for such metrics to be of vanishing Douglas curvature is provided. Moreover, many analytical examples are achieved by using a special kinds of metrical deformations called \(\beta \)-deformations.

On Asanov's Finsleroid–Finsler metrics as the solutions of a conformal rigidity problem

Finsleroid–Finsler metrics form an important class of singular (y-local) Finsler metrics. They were introduced by G.S. Asanov in 2006. As a special case of the general construction Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. The so-called Asanov's unicorns (Landsberg spaces that are not of Berwald type) were startlers in Finsler geometry although the existence of regular (y-global) Landsberg metrics that are not of Berwald type is an open problem up to this day. Independently of Asanov's works, the Finsleroid–Finsler metrics appeared in 2003 related to the invariance problem of the mixed Berwald curvature under conformal changes. In the special case of vanishing curvature the problem is due to M. Matsumoto: are there conformally equivalent non-Riemannian Berwald manifolds? Although the condition of the conformal invariance of the mixed Berwald curvature turned out to be too strong to admit non-homothetic conformal changes, the proof reduced the problem to the investigation of the so-called Asanov-type Finslerian metric functions called Finsleroid–Finsler metrics by Asanov three years later in 2006. Using some weakening of the conformal invariance of the mixed Berwald curvature Finsleroid–Finsler metrics with closed Finsleroid axis 1-forms are characterized as the solutions of a conformal rigidity problem in this paper. We are looking for (non-Riemannian) Finsler metrics admitting a (non-homothetic) conformal change such that the mixed curvature tensor of the Berwald connection contracted by the derivatives of the logarithmic scale function is invariant. We prove that any solution must be locally conformal to a Finsleroid–Finsler metric with closed Finsleroid axis 1-form. Conversely, a Finsleroid Finsler metric with closed Finsleroid axis 1-form admits a local conformal change satisfying the rigidity condition.

Lelong Numbers and Vector Bundles

We study Lelong numbers and integrability indices for S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^1$$\\end{document}-invariant singular metrics on vector bundles over the disk.

On the asymptotic behavior of Bergman kernels for positive line bundles

Let L be a positive line bundle on a projective complex manifold. We study the asymptotic behavior of Bergman kernels associated with the tensor powers L^p of L as p tends to infinity. The emphasis is the dependence of the uniform estimates on the positivity of the Chern form of the metric on L. This situation appears naturally when we approximate a semi-positive singular metric by smooth positively curved metrics.

Generalized Vaidya solutions and Misner-Sharp mass for n -dimensional massive gravity

Dynamical solutions are always of interest to people in gravity theories. We derive a series of generalized Vaidya solutions in the $n$-dimensional de Rham-Gabadadze-Tolley (dRGT) massive gravity with a singular reference metric. Similar to the case of the Einstein gravity, the generalized Vaidya solution can describe shining/absorbing stars. Moreover, we also find a more general Vaidya-like solution by introducing a more generic matter field than the pure radiation in the original Vaidya spacetime. As a result, the above generalized Vaidya solution is naturally included in this Vaidya-like solution as a special case. We investigate the thermodynamics for this Vaidya-like spacetime by using the unified first law, and present the generalized Misner-Sharp mass. Our results show that the generalized Minser-Sharp mass does exist in this spacetime. In addition, the usual Clausius relation $\delta Q= TdS$ holds on the apparent horizon, which implicates that the massive gravity is in a thermodynamic equilibrium state. We find that the work density vanishes for the generalized Vaidya solution, while it appears in the more general Vaidya-like solution. Furthermore, the covariant generalized Minser-Sharp mass in the $n$-dimensional de Rham-Gabadadze-Tolley massive gravity is also derived by taking a general metric ansatz into account.

Log canonical thresholds on group compactifications

We compute the log canonical thresholds of non-negatively curved singular hermitian metrics on ample linearized line bundles on bi-equivariant group compactifications of complex reductive groups. To this end, we associate to any such metric a convex function whose asymptotic behavior determines the log canonical threshold. As a consequence we obtain a formula for the alpha invariant of these line bundles, in terms of the polytope associated to the group compactification.

Approximations and examples of singular Hermitian metrics on vector bundles

We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose standard approximations are not Nakano semipositive. The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank 2 bundles on elliptic curves.

An observation on Asanov's Unicorn metrics

Finsleroid-Finsler metrics form an important class of singular (y-local) Finslerian metrics. They were introduced by G. S. Asanov in 2006. As a special case Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. These are called Unicorns [5]. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day. In this paper we prove that Asanov's Unicorns belong to the class of generalized Berwald manifolds. More precisely we prove the following theorems: a Finsleroid-Finsler space is a generalized Berwald space if and only if the Finsleroid charge is constant. Especially a Finsleroid-Finsler space is a Landsberg space if and only if it is a generalized Berwald manifold with a semi-symmetric compatible linear connection.

Ueda theory for compact curves with nodes

Let $C$ be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of $C$ when $C$ is non-singular or is a rational curve with a node. We propose an analogue of Ueda's theory for the case where $C$ admits nodes. As an application, we study singular Hermitian metrics with semi-positive curvature on the anti-canonical bundle of the blow-up of the projective plane at nine points in arbitrary position.

Equidistribution on Big Line Bundles with Singular Metrics for Moderate Measures

We establish an equidistribution theorem for the common zeros of random sections of high powers of several singular Hermitian big line bundles associated to moderate measures.

Classification theorems for solutions of higher order boundary conformally invariant problems, I

In this paper, we prove that nonnegative polyharmonic functions on the upper half space satisfying a conformally invariant nonlinear boundary condition have to be the “polynomials plus bubbles” form. The nonlinear problem is motivated by the recent studies of boundary GJMS operators and the Q-curvature in conformal geometry. The result implies that in the conformal class of the unit Euclidean ball there exist metrics with a single singular boundary point which have flat Q-curvature and constant boundary Q-curvature. Moreover, all of such metrics are classified. This phenomenon differs from that of boundary singular metrics which have flat scalar curvature and constant mean curvature, where the singular set contains at least two points. A crucial ingredient of the proof is developing an approach to separate the higher order linear effect and the boundary nonlinear effect so that the kernels of the nonlinear problem are captured.